=Paper=
{{Paper
|id=Vol-1901/paper24
|storemode=property
|title=Retinamorphic color Schrodinger metamedia
|pdfUrl=https://ceur-ws.org/Vol-1901/paper24.pdf
|volume=Vol-1901
|authors=Valeri Labunets,Ivan Artemov,Victor Chasovskikh,Ekaterina Ostheimer
}}
==Retinamorphic color Schrodinger metamedia ==
https://ceur-ws.org/Vol-1901/paper24.pdf
Retinamorphic color Schrödinger metamedia
V. Labunets1, I. Artemov1, V. Chasovskikh1, E. Ostheimer2
1
Ural State Forest Engineering University, 620100, Ekaterinburg, Russia
2
Capricat LLC, Pompano Beach, Florida, USA
Abstract
In this work, we use quantum color cellular automata to study pattern formation and image processing in quantum-diffusion Schrödinger
systems with triplet-valued (color-valued) diffusion coefficients. Triplet numbers have the real part and two imaginary parts (with two
imaginary units and , where 1 ). They form 3-D triplet algebra. Discretization of the Schrödinger equation gives quantum color
1 2 3
cellular automata with various triplet–valued physical parameters. The process of excitation in these media is described by the color
Schrödinger equations with the wave functions that have values in triplet algebras. The color Schrödinger metamedia can be used for creation
of the eye-prosthesis. The color metamedium suggested can serve as the prosthesis prototype for perception of the color images.
Keywords: Color Schrödinger equation; color Schrödinger transform; color metamedia; cellular automata; silicon eye; color image processing
1. Introduction
Diffusion and Schrödinger equations (linear and nonlinear) [1,2] and real-valued Gauss, complex-valued Fresnel and
Schrödinger transforms associated with them are important members of the family of methods for image processing, computer
vision, and computer graphics. Schrödinger transform of image as a new tool for image analysis was first given in [2]. Neural
networks and cellular automata (in form of a media) which are compatible with the theory of quantum mechanics and
demonstrate the particle-wave nature of information have been analyzed in [3-5]. The studying of processes in such metamedia
is very important for many branches of the system theory. There is no general theory of the metamedia yet, and every particular
example of similar media, usually provides us with the examples of new dynamic or self-organization types.
In this work, we apply quantum cellular automata to study pattern formation and image processing in quantum-diffusion
Schrödinger metamedia with triplet-valued diffusion coefficients. Triplet (color) numbers [6] contain one real and two imaginary
components with two hyper-imaginary units 1 and 2 and the following property 3 1: C r g 1 b 2 , where r , g , b are
real numbers. The numbers C r g 1 b 2 are called triplet or color numbers. They form a 3-D triplet (color) algebra
A3 ( ) A3 (R |1, 1, 2 ): C r g 1 b 2 | r, g, b R . If the diffusion coefficient in the Fourier diffusion or Planck’s
constant in the Schrödinger equations are a triplet number D r g 1 b 2 then both equations are turned into the color
Schrodinger equation. It describes the process of excitement in the so-called color Schrodinger metamedium with A 3 ( ) -valued
(color) wave function ( x , y , t ) r ( x , y , t ) g ( x , y , t ) 1 b ( x , y , t ) 2 | .
In this work, we study properties of the color Schrödinger excitable metamedium in the form of a cellular automaton. The
more detailed information about cellular automata can be found in [7]. The automaton's cells are located inside a 2D array. They
can perform basic operations with triple (color) numbers (in color algebra A 3 ( R | 1, 1 , 2 ) ). These cells are able to inform the
neighboring cells about their states. Such media possess large opportunities in processing of color images in comparison with the
ordinary diffusion media with the real-valued diffusion coefficient.
The rest of the paper is organized as follows: in Section 2, the object of the study (the color Schrödinger equation) is
described. In Section 3, a brief introduction to mathematical background (color algebra A 3 A 3 ( R | 1, 1 , 2 ) of triplet numbers
C r g 1 b 2 ) is given (subsection 3.1) in order to understand the concept behind the proposed method. In subsection 3.2, the
proposed method based on color Schrödinger equations is explained. Next, we defined Schrödinger transform of color image,
discussed its properties. In Section 4, the basic color metamedia (the color Schrödinger-Euclidean, color Schrödinger-
Minkowskian, color Schrödinger-Galilean and color Schrödinger-Yaglom) are devised and analyzed in detail. The simulation
result and algorithm complexity are demonstrated too. Finally, we gave our conclusion in Section 5.
2. The object of the study
In this work, we apply quantum cellular automata to study pattern formation and image processing in color quantum-
diffusion Schrödinger metamedia with triplet-valued diffusion coefficients:
( x , y , t ) 2 ( x , y , t ) 2 ( x , y , t ) (1)
D f x, y, t ,
t x 2 y 2
where (x, y,t) r (x, y, t) g (x, y,t) b (x, y,t)
1 2
is a color wave function that describes excitement of medium,
f x , y , t f r ( x , y , t ) f g ( x , y , t ) f b ( x , y , t )
1 2
is an exciting color source (input color signal) and D r g 1 b 2 is color-
valued diffusion coefficient. It describes the process of excitement in the so-called color Schrodinger metamedium with A 3 ( ) -
valued (color) wave function ( x, y, t ) . Discretization of the color Schrödinger equation gives a color quantum Schrödinger
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cellular automaton with various triple–valued physical parameters. Their microelectronic realizations appear to be a
programmable Schrodinger metamedia [8]. The main purpose of this work is the investigation of time evolution for color
Schrödinger metamedia in the form of quantum cellular automata with triplet diffusion coefficients. The automaton's cells are
located inside a 2D array. They can perform basic operations with triple (color) numbers (in color algebra A 3 ( R | 1, 1 , 2 ) ).
These cells are able to inform the neighboring cells about their states. Such media possess large opportunities in processing of
color images in comparison with the ordinary diffusion media with the real-valued diffusion coefficients. The latter media are
used for creation of the eye-prosthesis (so called the "silicon eye"). The medium suggested can serve as the prosthesis prototype
for perception of the color images [9-15].
3. Methods
3.1. Mathematical background. Triplet algebra
Let us consider the algebraic and geometric properties of the triplet algebra A 3 A 3 ( R | 1, 1 , 2 ) :
C r g1 b 2 | r, g, b R . The addition and product of two triplet numbers C1 (r1 g1 b1 2) and
C2 (r2 g2 b2 2) are given by [6]:
C1 C2 r1 g11 b1 2 r2 g21 b2 2 r1 r2 g1 g2 1 b1 b2 2 ,
C1 C2 ( r1 g 1 b1 2 ) ( r2 g 2 b 2 2 ) r1 r2 b1 g 2 g 1 b 2 g 1 r2 r1 g 2 b1 b 2 b1 r2 g 1 g 2 r1 b 2 2 .
It is useful to introduce the following triplet numbers elum : 1 /3 , E chr : (1 3 2 32 ) / 3,
2
where
3 exp i 2 / 3 . It is easy to check e 2
lum e lum , E 2
chr = E chr , e lum E chr E chr e lum 0 . Hence, elum , E chr are orthogonal idempotents
(projectors) and every triplet (color) number C r g b 2 can be represented in the form of the linear combination of a
"scalar" alum e lum and "complex" zchr Echr components C alum elum zchr Echr alum , zchr in the idempotent basis e lum , E chr , where
alum e lum = C e lum , zchr E chr C Echr , because
C elu (alu elu zch Ech ) elu alu elu2 zch Echelu alu elu =
,
C Ech (alu elu zch Ech ) Ech alu elu Ech Zch Ech
2
zch Ech .
We will call real numbers alum R the luminance numbers and complex numbers z chr C - the chromatic numbers. Obviously,
1 1 2 1 1 2
alum elum C elum (r g 1 b 2 ) ( r g b) ,
3 3
1 1 1 2 2 1 1 1 2 2
zchr Echr C Echr (r g 1 b 2 ) (r g1 b 2 ) .
3 3
Hence, alum r g b , z chr r g 1 b 2 r g b i 3
( g b ). In the new duplex representation two main arithmetic
2 2
operations have the simplest form:
C B alum elum zchr Echr blum elum wchr Echr alu blu elum zchr wchr Echr ,
C B alum elum zchr Echr blum elum wchr Echr alu blu elum zchr wchr Echr .
Consequently, a color algebra A3() is the direct sum of real R and complex C fields: A 3 R e lu C E ch R C . It
is known that every 2-D complex number z x iy can be represented geometrically by the modulus z x iy
z x2 y 2 and by the polar angle arctg x / y . The modulus is multiplicative and the polar angle is additive
upon the multiplication of ordinary complex numbers. The triplet numbers introduced in this section have the form
the variables r, g and b being real numbers. In a geometric representation, the triplet number
is represented by the point C (r , g , b), or as a 3-D vector with coordinates ( r , g , b ) in the 3-D color space
R 3c o l (see Fig. 1).
Let the point O be the point of the origin of the R, G, B axes, and T A c h will be the line, which contains the points with equal
coordinates r g b (it is called an achromatic diagonal). The luminance numbers a lum lie on this achromatic diagonal.
Also let M (alu ) be the plane r g b alum that is perpendicular to an achromatic axis T A c h ; this plane intersects it on a
range a lum from the point of origin O. It is called a chromatic plane. It contains chromatic numbers zchr .
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a) b)
r g b 2 in the form of a 3D vector C= r , g , b R 3col or as a point
Fig. 1. a) The geometrical representation of a triplet number C=
C= C ( r , g , b ) R 3col in 3-D color space R 3c o l . The geometrical characteristics have the following values: a lum r g b / 3,
d r2 g2 b2 , Rchr d2 alum
2
, chr arg zchr b) the color cube, it's achromatic diagonal and chromatic plane.
Fig. 2. The relation between an angle chr ( x , y , t ) and hue Hchr ( x, y, t ).
Obviously, a vector C= r g b 2 ( r , g , b ) can be described: 1) by the projection a lum of a line segment OC on a
line T A c h , i.e. by the luminance component and 2) by a complex number zchr in the chromatic plane. Besides, the absolute value
of this number appears to be the range z ch r from C ( r , g , b ) to this line, i.e. it describes the saturation (which is marked by the
symbol S chr z chr ) of a triplet number C r g b 2 and the azimuth angle chr arg z chr represents its color hue (which
we can mark by the symbol H chr chr arg z chr ) according to Fig. 2.
3.2. The generalized Schrödinger equation and cellular automata
Let a diffusion coefficient D r g 1 b 2 in the Schrödinger equation
( x , y , t ) 2 ( x , y , t ) 2 ( x , y , t ) (2)
( r g 1 b 2 ) f x, y , t ,
t x 2 y 2
be a triplet number, where (x, y,t) r (x, y,t) g (x, y,t)1 b (x, y,t)2 is a color wave function that describes excitement of medium,
f x , y , t f r ( x , y , t ) f g ( x , y , t ) 1 f b ( x , y , t ) 2 is an exciting color source (input color signal). Color wave function ( x, y, t )
describes time evolution of state ( x, y, t ) (in terms of triplet numbers) of a metamedium point with coordinate ( x , y ) . If
D Dcl r R is a real number and ( x , y , t ) r ( x, y , t ) then (2) is an ordinary diffusion (or heat) equation for the real
ordinary medium (we will call one as the Fourier-Gauss medium). If D iD qu C is an imaginary number and
( x , y , t ) cl ( x , y , t ) i qu ( x , y , t ) then (2) becomes an ordinary Schrödinger equation with the Plank's constant iD qu i / 2 m
for ordinary quantum Schrödinger medium. If D D cl iD qu A 2 ( R | i ) and ( x , y , t ) cl ( x , y , t ) i qu ( x , y , t ) then (2) is
bichromatic Schrödinger equation for bichromatic quantum Schrödinger metamedium [16]. It is a generalization of both
diffusion and Schrödinger metamedia.
In case of zero initial conditions, we can write the solution of (2) in the form of the Cauchy integral:
- x - y -
2 2
t
1 d .
( x , y, t )
- -
e
4 D t -
f , , d d (3)
2
0 2 D t -
This integral we will call the color Schrödinger transform (GST) of the initial image f ( x , y , t ). If
iD qu i / 2 m C A 2 ( R | i ), then GST is ordinary Schrödinger transform [1-5].
Let us introduce a 2-D regular lattice with nodes xn , ym , t k , where x n 1 x n h , y m 1 y m h and t k 1 t k . Here h
and are spaces between nodes on the space Z 2S p R 2 and time Z t R t lattices, respectively. For discrete Laplacian we use
the following approximation:
d 2 / dx 2 ( xn 1, ym , tk ) ( xn 1, ym , tk ) 2 ( xn , ym , tk ),
d 2 / dy 2 ( xn , ym 1, tk ) ( xn , ym 1, tk ) 2 ( xn , ym , tk ), (4)
d / dt ( xn , ym , tk 1) ( xn , ym , tk ).
2
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As a result, we get the 2-D discrete color Schrödinger equation
(xn , ym , tk 1) ( xn , ym , tk )
(5)
D ( xn 1, ym , tk ) (xn 1, ym , tk ) (xn , ym 1, tk ) ( xn , ym 1, tk ) 4(xn , ym , tk ).
Now, we give the definition of a 2-D “cellular space” (2-D regular lattice) in which the cellular automaton is defined. A regular
lattice Z 2Sp R 2Sp consists of a set of cells (elementary automata, or electrical circuits Au
ut ), which homogeneously cover a 2-D
Euclidean space. Each cell is labeled by its position A uut ( x n , y m ) A uut ( n , m ), n , m Z 2Sp
Regular, discrete, infinite network consisting of a large number of simple identical elements in the form of elementary
automata Auut ( n, m ) a copy of which will take place at each node ( n , m ) of the net is called the cellular automaton (see Fig.2
and Fig.3a in [16] ). Each so decorated note will be called a cell Auut ( n, m ) and will communicate with a finite number of other
, ) M, M(mn
cells Auut (i, k ) , which determine its neighborhood (i, k ) M ( m, n ) , geometrically uniform M(mn , )ZSp . The 2
neighborhood of the cell Auut ( n, m ) (including the cell itself or not, in accordance with convention) is the set of all the cells
Auut (i, k ) , (i, k ) M ( m, n ) of the network which will locally determine the evolution of Auut ( n, m ) . This local communication,
which is deterministic, uniform and synchronous determines a global evolution of the cellular automaton, along discrete time
steps t k 1 t k .
In the case of Z 2Sp , the classical neighborhoods are the von Neumann and Moore ones. They are known as the nearest
neighbors neighborhoods, and defined according to the usual norms and the associated distances. More precisely, for (i, j)ZSp
2
, || (i, j ) ||1 | i | | j | and || ( i , j ) || m ax | i |, | j | will denote 1 - and -norm respectively. Let 1 and be the associated
distances. Then Von Neumann and Moore neighborhoods (Fig.2) are M (m, n) : (i, k) | 1 (m, n),(i, k ) 1 and
M (m, n) : (i, k) | (m, n),(i, k) 1 , respectively. To each cell Auut ( n, m ) we assign an A2 (R | i) -valued state
(n, m, k ) ( xn , ym , tk ) (i.e., the media's excitement). The dynamics of the cellular automaton are determined by a local
transition rule, which specifies the new state ( n, m , k 1) ( xn , ym , tk 1 ) of a cell as a function of its interaction Von
Neumann neighborhood configuration, according to (5), i.e.,
( n , m , k 1) ( n , m , k ) D ( n 1, m , k ) ( n 1, m , k ) ( n , m 1, k ) ( n , m 1, k ) 4 ( n , m , k ) . (6)
This rule shows us the relation between a state ( n , n , k 1) of the cell Auut ( n, m ) at the current moment time k 1 and the
state ( n, m , k ) the same cell Auut ( n, m ) and the states of the four neighboring cells ( n 1, m , k ), ( n 1, m, k ),
( n, m 1, k ), ( n , m 1, k ) at the previous moment time k .
4. Results and Discussion
4.1. The Schrodinger-Euclidean metamedium
We can write the Schrödinger equation (2) in the idempotent basis e lum , E chr . Because
( x , y , t ) r ( x , y , t ) g ( x , y , t ) 1 b ( x , y , t ) 2 lum ( x , y , t ) e lum chr ( x, y , t ) E chr ,
f x , y , t f r ( x , y , t ) f g ( x , y , t ) 1 f b ( x , y , t ) 2 f lum ( x , y , t ) elum f chr ( x , y , t ) E chr ,
D r g 1 b 2 Dlum elum Dchr E chr ,
the equation (2) breaks down onto two equations:
a) b) c) d)
Fig. 3. The state of the color Schrodinger-Euclidean metamedium at moments a) tk 16 and b) tk 210 , when Dlum Schr , chr 0
and c) tk 13 and d) tk 120 , when Dlum Schr , chr 0.
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d d2 d2
lum ( x, y , t ) Dlum 2 lum ( x, y , t ) 2 lum ( x, y , t ) ,
dt dx dy
(7)
d d2 d2
chr ( x, y , t ) Dchr 2 chr ( x, y, t ) 2 chr ( x, y , t ) .
dt dx dy
one for a luminance and other - for a chromatic components.
i chr (x, y,t)
Obviously, chr (x, y,t) chr (x, y, t) e S(x, y,t)eiHchr (x, y,t) , where S ( x , y , t ) chr ( x , y , t ) , H chr ( x , y , t ) chr ( x , y , t )
are the saturation and the hue of a wave function, respectively. The relation between an angle chr ( x, y, t ) and a color hue
H chr ( x , y , t ) is shown on a Fig. 2. The first expression in (7) is the equation of a heat conduction with a real-valued diffusion
coefficient Dlum rD g D bD . It describes the time brightness evolution lum ( x , y , t ) of a wave function ( x, y, t ) . The second
g D bD 3
expression appears to be the Schrodinger equation with a complex diffusion coefficient Dchr rD i ( g D bD ). It
2 2
describes the time hue evolution chr ( x, y, t ) of the wave function.
For the modeling results representation we will use the cellular automaton, in which the cell's states are shown as color
pixels (as triplet numbers). The sum of four Dirac's delta-functions (red, green, white and blue) as an input signal f x , y , t . On
Fig. 3 these functions are represented as four points of the corresponding colors. Each figure consist of four parts: the top left
quarter shows the resulting RGB picture (i.e. presents wave function ( x, y, t ) in the RGB format), the top right part shows the
luminance component lum ( x , y , t ) of a color wave function, the bottom left one shows the saturation chr ( x , y , t ) S ( x , y , t )
and the last one represents the color tone chr ( x, y, t ) .
Initially we will consider time evolution of the Schrodinger-Euclidean metamedium for the "balanced" chromatic and
achromatic parameters D D lum e lum S chr e i H E chr where Dlum Schr , chr 0 , i.e. D D lu m e lum E ch r . In this case we
ch r
take the equal values of a diffusion coefficient's luminance and saturation, when the chromatic phase is equal to zero:
Dlum Schr , chr 0 . The results of a simulation for this case are shown on Fig. 3a ( tk 16 ) and Fig. 3b ( t k 120 ). Fig. 3c-d
shows the process of a color excitement's propagation in a color metamedium, which diffusion coefficient has the low value of
saturation ( Dlum Schr , chr 0 ). The achromatic components on all illustrations in this work are inverted to reduce the amount
of dark colors for a better visual perception of pictures. Therefore, the darker colors mean higher values of excitement. Note that
chromatic parts of all spots are spreading slower than achromatic ones: the size of spots in the top right quad (excitement's
luminance representation) is bigger than in the bottom left one (excitement's saturation representation).
Results that are more interesting can be obtained when we increase the value of a color hue chr of a diffusion coefficient.
As an input signal we use a single red-colored Dirac's delta-function that is affecting the central point of a cellular automaton.
For a comparison, it is important to see the excitement of cellular automaton with a zero color hue chr 0 (when
Dlum S chr 0.11 ). The results are presented on Fig. 4. This picture shows only resulting RGB images (the top part) and cells'
chromatic phases (the bottom part). Also, note the Fig 5.
a)
b)
c)
Fig. 4. The state of time evolution of the color Schrodinger-Euclidean metamedium at moments tk 0, 10, 70, 160
a ) chr 0 b chr c ) chr 0 ).
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4.2. The Schrodinger-Yaglom color metamedia
The chromatic plane, in which D ch r D ch r e i ch r
S ch r e i H c h r lays, appears to be a classic complex algebra with i2 1 .
It is interesting to study a color metamedium with a chromatic plane in the form of another two complex algebra with i2 1
and i02 0 [17] , i.e. with the following chromatic components:
Dchr Dchr ei ch Sch ei Hch and Dchr Dchr ei0ch Sch ei0Hch .
We will call such media the color Schrodinger-Yaglom metamedia. The Fig. 6a contains excitements for color Schrodinger-
Galilean metamedium at the moment of time t k 128 (for the same input signal as in the previous case) for different values of
the hue of the diffusion coefficient ( chr 5 , 40, 60 ). As we can see on a Fig. 6b, the further increase of the hue (for
diffusion coefficient) chr 70 , 89, 90 leads to the fast concentration and contraction of a phase circle in the middle of
the bottom right square. In addition, our red-colored initial point completely turns into a spot with a pearl halo when the hue of
the coefficient D reaches chr 90 . In addition, we should mention that values chr arg{Dchr } do not produce any new
phenomena because of the periodic nature of trigonometric functions. Indeed, the color excitement with arg{ Dchr } chr 90 0
turns out to be the inverted (by a color tone) excitement of a metamedium with arg{Dchr } chr 90 (see the Fig. 7a that is
quite similar to Fig. 6b).
The example of the excitement of the Schrödinger-Minkowskian metamedium with a chromatic component of a diffusion
i chr
coefficient in the form of a double number Dchr Dchr e Schr ei Hch is shown on a Fig. 7b.
Fig. 5. The typical form of an excitement for Schrodinger-Euclidean metamedium under the impact of an input white Dirac's delta-impulse.
a)
b)
Fig. 6. The state of time evolution of the color Schrodinger-Galilean metamedium ( i 0 ) at the moment tk 128 : a) chr 5 , 40, 60 ,
2
b) chr 70 , 89, 90
4.3. The excitement of the color Schrödinger metamedium by a moving source
Let the excitement function f x , y , t in equation (2) be the Dirac delta-function that is moving on the circle with a
radius R and the center at the point x0 , y0 . The source has an angular velocity :
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x0 x(t) y0 y(t) R2. It means that we have a moving quantum
2 2
f x , y , t x0 R cos( t ), y 0 R sin( t ) , where
particle in a color metamedium. Firstly, we will research the color Schrodinger-Euclidean metamedium with the chromatic
i chr
component in the form of a classical complex number Dchr Dchr e Schr ei Hchr that has a relatively low chromatic phase
value chr of D chr . The Fig.8a-b demonstrates the results of modeling for this case.
a)
b)
Fig. 7. a) The excitement of a color Schrödinger-Galilean metamedium ( i 0 ) at the moment tk 128 for the diffusion coefficients with hues
2
chr 100,130,175 ; b) The excitement of the color Schrödinger-Minkowskian metamedium ( i 2 1 ) at the moment tk 128 for the diffusion
coefficients with hues chr 100,130,175
a) chr 5 b) chr 60 c) chr 74 d) chr 85
Fig. 8. The excitement of the color Schrödinger-Euclidean metamedium by a particle moving on a circular trajectory ( t k 128 ).
a) chr 70 b) chr 90
Fig. 9. The excitement of the Schrödinger-Galilean metamedium by a particle moving on a circle ( t k 100, i02 0 ).
When the chromatic angle chr values are small (low color tone) then D ch r 's excitement fluctuation components, that are
perpendicular to the movement trajectory, are almost absent. We only can see the parts of fluctuations that exist along the
trajectory. When values of the chromatic angle chr (hue) are being increased, we can observe the excitement's fluctuations that
are perpendicular to the trajectory of a movement. Also the interference of a "tail" and "head" parts becomes visible (see Fig. 8c-
d).
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Different results can be obtained for color media with a chromatic component in the form of a double Dchr S chr e i and a 0 chr
dual Dchr S chr e i number. For example, when we use a dual number Dchr S chr e i the alteration of chr arg{Dchr } leads
0 chr 0 chr
to some interesting and even more unusual consequences. The Fig. 9 shows the pictures of an excitement at the moment
t k 100 for the quite high values of a phase chr (the picture of an excitement changes weakly for the wide range of chr 's
values). It can be seen that in the case of a Schrödinger-Galilean metamedium, the growth of a Dchr 's phase causes the increase
of a violet and pearl color amounts (on condition that the moving particle has a red color). Particle trail's halo on the right part of
Fig. 9 is quite bright, but there are no cells with high lightness and saturation parameter values in the investigated area. It is
caused by irregular laws of the behavior of the chromatic component for this metamedia type.
a) b)
Fig. 10. The interference of four excitements at the moment tk 128 in a) the Schrodinger-Galilean metamedium ( i
2
0 ) and b) the Schrödinger-
Minkowskian metamedium ( i
2
1 ) metamedia ( chr 50 in both cases).
4.4. The interference of excitements in the color Schrödinger metamedia
The process of excitement's interference in Schrödinger-Euclidean metamedia has a classic character. The results of a
simulation for Schrodinger-Galilean ( i 2 0 ) and Schrödinger-Minkowskian ( i 2 1 ) metamedia are shown on Fig. 10a and
Fig. 10b, respectively. It can be seen on Fig. 10a that in the Schrödinger-Galilean metamedium the collision of different-colored
excitements produces unusual rays in the areas where an occlusion happened. There are no such phenomena in the Schrodinger-
Euclidean and in the Schrödinger-Minkowskian metamedia.
a) b)
Fig. 11. a) The excitement function (input image) f ( x, y, t0 0) of the color Schrödinger-Euclid metamedium at the initial moment t 0 0 ; b) The
excitement of this metamedium at the moment tk 128 . The metamedium has broken the image onto the areas of uniformity with respect to brightness and
hue at this time.
4.5. Some applications of the color Schrödinger metamedia
Let the excitement function f ( x , y , t 0 0) f r ( x , y , 0) f g ( x , y , 0) f b ( x , y , 0) 2 f lum ( x , y , 0) e lum f chr ( x , y , 0) E chr in
(2) represents a color RGB image at the moment t0 0. Then the color wave function
( x , y , t ) r ( x , y , t ) g ( x , y , t ) b ( x , y , t ) 2 lu m ( x , y , t ) e lu m chr ( x , y , t ) E chr (8)
shows us the time evolution of initial image f ( x, y , t0 0) ( x, y, 0) . As an example of such image, we take a flower in an
RGB format (see Fig. 11b, top left quarter). The luminance component lum ( x , y , t ) of wave function (8) represented in the
bottom left part of Fig. 11b, the saturation component chr ( x , y , t ) - in the top right quarter and the hue ( x, y, t ) arg chr ( x, y, t )
- in the bottom right part.
One of the most important tasks in the digital processing of color images [18] is the distinguishing of image's parts,
where some of its components have uniform values. It is the uniformity areas detection, for example, we can detect the areas
with a similar brightness, saturation or color tone, etc. Usually one has to perform such operation before starting the image
segmentation by some parameter. It turns out that color Schrödinger metamedia are able to implement such operations. Fig. 11b
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shows the excitement of a Schrödinger-Euclidean metamedia at the moment t 128 after an impact that is represented as an
image, which was described previously. It is easy to see that by this time the metamedia has broken the initial image onto areas
of uniformity by luminance and by color tone. Fig. 12 and Fig. 13 shows the excitements of the Schrödinger-Galilean and
Schrödinger-Minkowskian metamedia at the moments t k 0, 32, 64,128,160 and tk 0,84 , respectively, after an input impact in
the form of an initial image.
5. Conclusion
The metamedia with triplet (color) diffusion coefficients were first studied. Their laws of functioning are described by color
Schrodinger equations. Simulation of these equations in the form of quantum cellular automata was considered. The results of
modeling that were shown in this work demonstrate the complex character of the time evolution of such metamedia. Our future
work will be focused on using commutative and Clifford algebras for hyperspectral image processing and pattern recognition.
Fig. 12. The excitement of a Schrödinger-Galilean metamedia at moments of time tk 0,32,64,128,160 when an input signal had the form of the
initial image in the top left quarter.
a) b)
Fig. 13. The excitement of a Schrödinger-Minkowskian at the moment of time tk 0,84 when an input signal had the form of an image.
Acknowledgements
This work was supported by grants the RFBR № 17-07-00886, № 17-29-03369 and by Ural State Forest University
Engineering’s Center of Excellence in “Quantum and Classical Information Technologies for Remote Sensing Systems”.
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